Space Curves Research Articles

Parallel Software to Offset the Cost of Higher Precision

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

Spinning black holes as cosmic string factories

We consider the evolution of a cosmic string loop that is captured by a much more massive and compact black hole. We show that after several reconnections that produce ejections of smaller loops, the loop that remains bound to the black hole moves on a nearly-periodic non-self-intersecting trajectory, "the orbit". The orbit evolves due to an energy and angular momentum exchange between the loop and the spinning black hole. We show that such evolution is mathematically equivalent to a certain continuous deformation of an auxiliary closed curve in a 3-dimensional space; for zero black-hole spin this deformation is curve-shortening that has been extensively studied by mathematicians. The evolution features competing effects of loop growth by the superradiant extraction of the black-hole spin energy, and loop decay by the friction of the moving string against the horizon. A self-intersection of an auxiliary curve corresponds to a capture by the black hole of a new string segment and thus an addition of a new captured loop. Possible asymptotic states of such evolution are shown to be strong emitters of gravitational waves. Whether reconnections prevent reaching the asymptotic states remains to be explored. Additionally, the orbit's shape also evolves due to an emission of gravitational waves, and a recoil of the black hole that changes the orbit and likely leads to self-intersections. We argue that for a significant range of the dimensionless tension $\mu$, string loops are captured by supermassive black holes at the centers of galaxies. This strongly motivates further study of interaction between string loops and black holes, especially the influence of this process on the black hole spindown and on the production of gravitational waves by strings created in galactic nuclei. We also discuss potential loop captures by primordial black holes.

The ideal lines on the phase diagrams of liquids in 2D space

We use the numerical simulations for several model systems to investigate the behavior of the ideal contours and their correlations with the liquid-gas coexistence curve in 2D space. These contours are the line of the unit compressibility factor (Zeno line) and the line of ideal enthalpy (H line). Earlier, in 3D space several similarity relations were found for these lines. For various 3D real and model systems they are the straight lines along all density-temperature plane. The deviations from the linear form occur in cases when the interaction potentials are relatively too short-ranged or too long-ranged. However, in all cases, independently of its form, the Zeno line is the tangent to the liquid binodal branch at relatively low temperatures. These properties were confirmed in 2D cases as well. But one more relation connecting the parameters of the critical point and the Zeno line parameters appears to be different in 2D and 3D cases.

Motion of general non‐null curve in Minkowski 3‐spaces associated with Landau‐Lifshitz equation

In that paper, firstly, we get two additional different non‐null space curve evolutions in Minkowski 3‐space by considering Landau–Lifshitz (LL) equation where we identify the spin vector with the binormal vector and the normal vector of these curves, respectively. Then, we obtain some links for constructing the moving non‐null space curves by using the integrable LL equation. Finally, we give as an application the exact solution of the moving non‐null curve evolutions obtained by taking the spin vector, the normal vector of the curve, and we showed graphically that these solutions are wave solutions.

The Mathematic Model Study of Optimum Align Method in Compass

In this paper, based on the study of the characteristics of horizontal well designed by optimum align method in compass software, we speculated that, the optimum align method is one kind of space circular arc method. Therefore, we built a horizontal well track design model based on spatial arc curve optimization alignment hypothesis. The orbit design problem of horizontal wells is transformed into the problem solving of multiple linear equations with trigonometric functions. If the kick off point, the target point and the dog leg degree of the first and second inclined sections are determined, the solution of the track parameters of the horizontal well is unique, and the track parameters of the horizontal well can be calculated accurately. The model result is the same as the optimum align method in compass program. Therefore, the algorithm essence of Compass Optimum Align design method is to solve the problem of multiple linear equations based on the optimal alignment of arc curves in space.

Design Method and Strength Verification for Wind Power Generators Based on Line Gear Increasers

Wind power generation is widely considered as one of the most potential solutions to the energy crisis. Common wind generators in the industry are usually extremely huge. However, these huge wind generators are not decently available for remote places like border posts or observation stations. To maximize electricity production, wind generators of compact sizes are desired. This paper presents the design method and strength verification for a wind power generator based on line gear increasers. The line gear is a new type of gear that meshes between a pair of space curves. Based on the theory of line gear, a new design method of gear increaser is proposed. The space curve shape design, the meshing stress analysis, and the strength verification are carried out for the line gears. This paper can provide reference for the miniaturization design of wind power generators.

Deformed contour segment matching for multi-source images

Robust and accurate multi-source matching is a difficult task due to significant nonlinear radiometric differences, background clutter, and geometric deformation in corresponding regions. Motivated by these existing problems, a discriminating yet robust combined descriptor for multi-source image matching, called deformed contour segment similarity (DCSS), is proposed in this work. First, the proposed DCSS, which is constructed by histogram of the combined contour features rather than the commonly used corner point and gradient, presents the accurate correspondence between image pairs and improves the descriptive ability to radiometric differences. Second, the deformed curve is presented via a finite-dimensional matrix Lie group to determine the similarity metric with an explicit geodesic solution. The geodesic distance, which indicates the nearest distance between curves in fluid space, is defined as the weight coefficient of the constructed histogram to enhance the robustness of the descriptor. The proposed algorithm utilizes the holistic contour information for the scoring and ranking of the shape similarity hypothesis, which can effectively reduce the influence of partially missing contours. Finally, a precise bilateral matching rule is used to perform the matching between the corresponding contour segments. Some experiments are carried out on various infrared-visible image data sets. The results demonstrate that the proposed DCSS achieves more robust and accurate matching performance than many popular multi-source image matching methods.

Certified numerical algorithm for isolating the singularities of the plane projection of generic smooth space curves

Isolating the singularities of a plane curve is the first step towards computing its topology. For this, numerical methods are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. This type of curve appears naturally in robotics applications and scientific visualization. In this setting, we show that the singularities can be encoded by a regular square system whose solutions can be isolated with certified numerical methods. Our analysis is conditioned by assumptions that we prove to be generic using transversality theory. We also provide a semi-algorithm to check their validity. Finally, we present experiments, some of which are not reachable by other methods, and discuss the efficiency of our method.

A Gravity-Referenced Moving Frame for Vehicle Path Following Applications in 3D

Moving path frames assigned to spatial curves are commonly used in the development of motion control laws for autonomous vehicles. This work presents the Gravity Normal frame, a novel navigation reference frame developed specifically for autonomous vehicle applications. This moving path frame incorporates the knowledge that many autonomous vehicles operate in a gravitational field, and control strategies must account for this. Given a curve in space that represents a desired trajectory, the proposed strategy generates a navigation frame that is well defined regardless of path curvature, while guaranteeing the normal vector is always normal to gravity and hence constrained to the horizontal plane, regardless of path torsion. Due to these characteristics, the Gravity Normal path frame is ideally suited for vehicles with distinct longitudinal and lateral dynamics since the resulting cross-track errors have a precise physical interpretation. The properties of the navigation frame are derived, and its usefulness is showcased through simulation. Finally, its applicability is demonstrated with flight experiments on a fixed-wing unmanned aerial vehicle.

Optimal planar facility location with dense demands along a curve

This paper studies a multi-facility location problem with demand distributed continuously along a curve (MFLCD) in a planar space. We first model a single facility location problem in this setting. When the curve is approximated by a collection of line segments, an exact algorithm is proposed to solve the problem. Then we extend the model to the multi-facility case and propose an “alternate location and allocation” (ALA) based algorithm. Extensive computational experiments are conducted to justify the proposed models and algorithms. We show that using the proposed model to deal with continuous demand directly is much more effective than using other models with discrete demand generated by demand aggregation. As a by-product, some sufficient conditions that ensure an error-free demand aggregation from continuous demand to discrete demand are provided. Computational results are in support of the solution quality and computational efficiency of the proposed ALA-based algorithm for solving MFLCD problems. Our sensitivity analysis also offers some insights and suggestions.

Spectral curves of hyperbolic monopoles from ADHM

Magnetic monopoles in hyperbolic space are in correspondence with certain algebraic curves in mini-twistor space, known as spectral curves, which are in turn in correspondence with rational maps between Riemann spheres. Hyperbolic monopoles correspond to circle-invariant Yang–Mills instantons, with an identification of the monopole and instanton numbers, providing the curvature of hyperbolic space is tuned to a value specified by the asymptotic magnitude of the Higgs field. In previous work, constraints on ADHM instanton data have been identified that provide a non-canonical realization of the circle symmetry that preserves the standard action of rotations in the ball model of hyperbolic space. Here formulae are presented for the spectral curve and the rational map of a hyperbolic monopole in terms of its constrained ADHM matrix. This extends earlier results that apply only to the subclass of instantons of JNR type. The formulae are applied to obtain new explicit examples of spectral curves that are beyond the JNR class.

Space Curves and Solitons of the KP Hierarchy. I. The l-th Generalized KdV Hierarchy

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\dots, lm+k\rangle$, where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.

The T-function of a parametric curve

In this paper, we introduce the T-function, T(s), which is a polynomial defined by means of a univariate resultant constructed from a given parametrization of an algebraic space curve . It is shown that , where are polynomials (the fibre functions) whose roots are the fibre of the ordinary singularities of multiplicity . Therefore, a complete classification of the singularities of , via the factorization of a resultant, is obtained.

Geometrical Formalism for Dynamically Corrected Gates in Multiqubit Systems

The ability to perform gates in multiqubit systems that are robust to noise is of crucial importance for the advancement of quantum information technologies. However, finding control pulses that cancel noise while performing a gate is made difficult by the intractability of the time-dependent Schrodinger equation, especially in multiqubit systems. Here, we show that this issue can be sidestepped by using a formalism in which the cumulative error during a gate is represented geometrically as a curve in a multi-dimensional Euclidean space. Cancellation of noise errors to leading order corresponds to closure of the curve, a condition that can be satisfied without solving the Schrodinger equation. We develop and uncover general properties of this geometric formalism, and derive a recursion relation that maps control fields to curvatures for Hamiltonians of arbitrary dimension. We demonstrate examples by using the geometric method to design dynamically corrected gates for a class of two-qubit Hamiltonians that is relevant for both superconducting transmon qubits and semiconductor spin qubits. We propose this geometric formalism as a general technique for pulse-induced error suppression in quantum computing gate operations.

Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi–Cartan–Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only nontrivial example of a totally biharmonic surface appears in the product space [Formula: see text].

Morgan-Voyce Polynomial Approach for Quaternionic Space Curves of Constant Width

Abstract The curves of constant width are special curves used in engineering, architecture and technology. In the literature, these curves are considered according to different roofs in different spaces and some integral characterizations of these curves are obtained. However, in order to examine the geometric properties of curves of constant width, more than characterization is required. In this study, firstly differential equations characterizing quaternionic space curves of constant width are obtained. Then, the approximate solutions of the differential equations obtained are calculated by the Morgan-Voyce polynomial approach.The geometric properties of this curve type are examined with the help of these solutions.

Metalens for Generating a Customized Vectorial Focal Curve.

Three-dimensional (3D) light fields with spatially inhomogeneous polarization and intensity distributions play an increasingly important role in photonics due to their peculiar optical features and extra degrees of freedom for carrying information. However, it is very challenging to simultaneously control the intensity profile and polarization profile in an arbitrary manner. Here we experimentally demonstrate a metalens that can focus light into an arbitrarily shaped focal curve with a predefined polarization distribution. The efficacy of this approach is exemplified through the demonstration of focused curves in 3D space ranging from simple shapes such as a circle to topologically nontrivial objects such as a 3D knot with controlled local polarization states. This powerful control of the light field would be technically challenging with their conventional counterparts. Our demonstration may find applications in beam engineering and integration optics.

A Modified DLS Scheme With Controlled Cyclic Solution for Inverse Kinematics in Redundant Robots

Redundancy in robotic manipulators has many advantages. It is successfully used to achieve better dexterity, and to avoid obstacles, singularities, or the kinematic limitations. However, redundancy makes the inverse kinematics (IK) problem harder to solve. The damped least squares (DLS) is a powerful method for calculating the IK of redundant robots, but it suffers from noncyclicity issue, where a closed curve motion in the Cartesian space of the end-effector (EEF) does not map into a closed curve in the joint space. This results in nonrepetitive motion in the joint space, even though the EEF motion is repetitive. In this article, we present a solution for the noncyclicity problem in the DLS method. The proposed scheme was successfully tested both in simulation (9 DoF robot) and on a real robotic manipulator (7 DoF robot).

Calculating the polarization in bipartite lattice models: Application to an extended Su-Schrieffer-Heeger model

We address the question of different representation of Bloch states for lattices with a basis, with a focus on topological systems. The representations differ in the relative phase of the Wannier functions corresponding to the diffferent basis members. We show that the phase can be chosen in such a way that the Wannier functions for the different sites in the basis both become eigenstates of the position operator in a particular band. A key step in showing this is the extension of the Brillouin zone. When the distance between sites within a unit cell is a rational number, $p/q$, the Brillouin extends by a factor of $q$. For irrational numbers, the Brillouin zone extends to infinity. In the case of rational distance, $p/q$, the Berry phase "lives" on a cyclic curve in the parameter space of the Hamiltonian, on the Brillouin zone extended by a factor of $q$. For irrational distances the most stable way to calculate the polarization is to approximate the distance as a rational sequence, and use the formulas derived here for rational numbers. The use of different bases are related to unitary transformations of the Hamiltonian, as such, the phase diagrams of topological systems are not altered, but each phase can acquire different topological characteristics when the basis is changed. In the example we use, an extended Su-Schrieffer-Heeger model, the use of the diagonal basis leads to toroidal knots in the Hamiltonian space, whose winding numbers give the polarization.

Some characterizations of rectifying curves in the 3-dimensional hyperbolic space $\mathbb H^{3}(-r)$

In this paper, we study the geometry of rectifying curves in the 3-dimensional hyperbolic space $\mathbb H^{3}(-r)$. Further we obtain the distance function in terms of arc length when the rectifying curve lying in the upper half plane. Then we find the distance function and also give the general equations of the curvature and torsion of rectifying general helices in $\mathbb{H}^3(-r)$.